Given equations,

$3x-2y+z=0$

$\lambda x-14y+15z=0$

$x+2y-3z=0$

If homogeneous system of linear equations has non trivial solution then determinant is zero.

Therefore we can say that, $∆=0 \Rightarrow \left[\begin{array}{ccc}3& -2& 1\\ \lambda & -14& 5\\ 1& 2& -3\end{array}\right]=0$

$\Rightarrow 3(42-30)+2(-3\lambda -15)+1(2\lambda +14)=0$

$\Rightarrow 20-4\lambda =0$

$\Rightarrow \lambda =5$